First integrals, integrating factors and λ-symmetries of second-order differential equations

2009 ◽  
Vol 42 (36) ◽  
pp. 365207 ◽  
Author(s):  
C Muriel ◽  
J L Romero
Keyword(s):  
Differential Equations ◽  
First Integrals ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Integrating Factors

scholarly journals Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

2020 ◽  
Vol 16 (4) ◽  
pp. 637-650
Author(s):  
P. Guha ◽  
◽  
S. Garai ◽  
A.G. Choudhury ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
First Integral ◽  
Lax Pair ◽  
First Integrals ◽  
Second Order ◽  
Lax Representation ◽  
Lax Pairs ◽  
Second Order Differential Equations ◽  
Autonomous Version ◽  
Autonomous Differential Equations

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

scholarly journals Interplay of symmetries, null forms, Darboux polynomials, integrating factors and Jacobi multipliers in integrable second-order differential equations

2014 ◽  
Vol 470 (2163) ◽  
pp. 20130656 ◽  
Author(s):  
R. Mohanasubha ◽  
V. K. Chandrasekar ◽  
M. Senthilvelan ◽  
M. Lakshmanan
Keyword(s):  
Differential Equations ◽  
Integrable Systems ◽  
Contemporary Literature ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Lie Point Symmetries ◽  
Darboux Polynomials ◽  
Emden Equation ◽  
Second Order Differential Equations ◽  
Integrating Factors

In this work, we establish a connection between the extended Prelle–Singer procedure with five other analytical methods which are widely used to identify integrable systems in the contemporary literature, especially for second-order nonlinear ordinary differential equations (ODEs). By synthesizing these methods, we bring out the interplay between Lie point symmetries, λ -symmetries, adjoint symmetries, null-forms, Darboux polynomials, integrating factors and Jacobi last multiplier in identifying the integrable systems described by second-order ODEs. We also give new perspectives to the extended Prelle–Singer procedure developed by us. We illustrate these subtle connections with the modified Emden equation as a suitable example.

scholarly journals On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations

2005 ◽  
Vol 461 (2060) ◽  
pp. 2451-2477 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Free Particle ◽  
First Integrals ◽  
Second Order ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Integrals Of Motion ◽  
Integrating Factors

A method for finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle–Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the PS method. We also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second-order nonlinear ordinary differential equations to free particle equations. We illustrate the theory with several potentially important examples and show that our procedure is widely applicable.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

scholarly journals New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Osama Moaaz ◽  
Choonkil Park ◽  
Elmetwally M. Elabbasy ◽  
Waed Muhsin
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
The Other ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.

scholarly journals Fixed point theorems via Generalized WF-Contractions with applications

SeMA Journal ◽  
2021 ◽  
Author(s):  
Rosana Rodríguez-López ◽  
Rakesh Tiwari
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
New Class ◽  
Second Order Differential Equations ◽  
Fixed Point Result

AbstractThe aim of this paper is to introduce a new class of mixed contractions which allow to revise and generalize some results obtained in [6] by R. Gubran, W. M. Alfaqih and M. Imdad. We also provide an example corresponding to this class of mappings and show how the new fixed point result relates to the above-mentioned result in [6]. Further, we present an application to the solvability of a two-point boundary value problem for second order differential equations.

Sign in / Sign up

Export Citation Format

Share Document